Finitely generated quadratic modules
∗Yuriy A. Drozd
Yuriy Drozd: Max-Plank-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany, and Kiev Taras Shevchenko University, De- partment of Mechanics and Mathematics, Volodimirska 64, 01033 Kiev, Ukraine;
e-mail: [email protected]
Mailing address:
Popudrenko 22/14, ap.19 02100 Kiev, Ukraine
∗This research was partially supported by Grant UM1-327 of CRDF and Ukrainian Government
Abstract. A complete description of finitely generated quadratic modules is given in terms of their projective presentations as well as of generators and relations. The main tool is the reduction of this description to a sort of “matrix problem.”
1. Introduction
Polynomial functors were introduced by Whitehead [15], Eilenberg and MacLane [13], and since then they have proved their essential role in a lot of problems of algebraic topology. In particular, their simplest case, quadratic functors, is widely used in homotopy theory (cf., e.g., [15, 2, 3]). They provide natural homotopy invariants of polyhedra [2], and someties these invariants are even enough for a complete classifica- tion of homotopy types [5]. Hence, a description of such functors seems an interesting and rather important problem. Fortunately, it is indeed quite possible, at least for the case of “right continuous” functors, de- termined by their values on free abelian groups, or, the same,quadratic modules in the sense of [2, 3]. Note that till now such a description has only been given for quadratic modules of “cyclic type” [2]. It turns out that the classification of quadratic modules is a special case of the problem considered by the author in [11], hence, can be reduced to the representations of the so called “bunches of chains” in the sense of [7]. In this paper we present such a reduction and deduce from [7] a complete description of finitely generated quadratic modules. Namely, after general definitions given in Section 2, we prove in Section 3 that the classification of quadratic modules should indeed be done locally, i.e., it is enough to consider their 2-adic localizations. Section 4 is the crucial one: it relates quadratic modules to the representations of a certain bunch of chains, while in Section 5 a description of quadratic modules is given via their projective presentations. In Section 6 we rewrite it in terms of generators and relations and give some corollaries of this description: projective resolutions of quadratic modules, their torsion free parts, etc.
We add an Appendix devoted to the representations of bunches of chains, where we reformulate and rearrange a trifle the results of [7]
taking into account the specific of our case (we need not but chains, while in [7] also semi-chains are considered). The answer is given in combinatorial frames of “strings and bands,” well-known to the ex- perts in the representation theory of finite-dimensional algebras. One may suppose that the relations between this theory and some prob- lems arising from topology should be rather wide and fruitful (cf., e.g., [4, 5]).
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2. Generalities
We remind some definitions and examples related to quadratic func- tors and quadratic modules. For the backgrounds we refer to the book [3]. We denote by Ab (resp., ab) the category of abelian groups (resp., that of finitely generated ones).
A quadratic functor is a functor F : Ab → Ab such that, for ev- ery two objects A, B, the function ˜F : Hom(A, B)×Hom(A, B) → Hom(A, B) defined as ˜F(a, b) = F(a +b) − F(a) − F(b) is bilin- ear. In this case, this function can be prolonged to a bilinear func- tor ˜F : Ab → Ab such that, for every A, B, one has: F(A⊕B) ' F(A)⊕F(B)⊕F˜(A, B) . Let F1 =F(Z) , F2 = ˜F(Z,Z) ; H :F1 →F2 be the composition F(Z)→F(Z⊕Z)→F˜(Z,Z) , where the first map- ping is induced by the diagonal embedding Z→Z⊕Z, while the second one is the projection onto the direct summand; at last, P : F2 → F1 be the composition ˜F(Z,Z) → F(Z ⊕Z) → F(Z) , where the first mapping is the embedding of the direct summand, while the second one is induced by the addition mapping Z⊕Z → Z. Then one has:
P HP = 2P and HP H = 2H, whence the following definition:
Definition 2.1. Aquadratic Z-module(or simply, aquadratic module) is a quadruple M = (M1, M2, H, P) , where M1, M2 are abelian groups and H : M1 → M2, P : M2 → M1 are homomorphisms such that P HP = 2P, HP H = 2H.
Conversely, each quadratic module M defines a quadratic functor M, called thequadratic tensor product: the group MA is generated by the symbols ma and n[a, b] , where m ∈M1, n∈M2, a, b∈A, subject to the following relations:
m(a+b) = ma+mb+Hm[a, b], (m+m0)a=ma+m0a ,
n[a, b] is 3-linear, n[a, a] =P(n)a ,
n[a, b] = [b, a](HP −1)a .
It is known [6] that the quadratic tensor product is right continuous, i.e., commutes with cokernels and direct limits, and every right contin- uous quadratic functor is isomorphic to M for a quadratic module M (determined up to isomorphism). Thus, the category of quadratic modules is equivalent to that of right continuous quadratic functors.
Remark. Certainly, a right continuous functor is completely defined by its values on the full subcategory fab of finitely generated free
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abelian groups. Thus, the category of quadratic modules is equivalent to that of quadratic functors fab→Ab. Just in the same way, one can identify the category of contravariant quadratic functors fab◦ → Ab (or, equivalently, that of contravariant left continuous quadratic func- tors Ab◦ →Ab) with the category dual to that of quadratic modules:
a quadratic module M corresponds to the “quadratic Hom-functor” HOM( , M) [3, Definition 6.13.14].
Quadratic modules can be considered as modules over a special ring A, for which we need a more explicit construction than that in [3].
Namely, we define A as the subring of the direct product Z×Z× Mat(2,Z) consisting of all triples
a, b,
c1 2c2 c3 c4
such that a≡c1 (mod 2) and b ≡c4 (mod 2). Let
e1 =
1,0, 1 0
0 0
, e2 =
0,1, 0 0
0 1
, h=
0,0,
0 0 1 0
, p=
0,0, 0 2
0 0
.
To each A-module M one associates the quadratic module (e1M, e2M, hM, pM) , where hM and pM denote, respectively, the multiplication by h and by p in the module M. Conversely, each quadratic module (M1, M2, H, P) gives rise to an A-module M such that M =M1⊕M2 as a group and
a, b,
a+ 2c1 2c2 c3 b+ 2c4
m1 m2
=
am1+c2P m2+c1P Hm1 bm2+c3Hm1+c4HP m2
. Hence, the category QM of quadratic modules is equivalent to the category A-Mod of (left) A-modules. Moreover, this correspondence maps finitely generated A-modules to finitely generated quadratic mod- ules (i.e., such that both M1 and M2 are finitely generated groups) and vice versa. The advantage of this realization of the ring A is that it is anorder in the semi-simple algebra Q×Q×Mat(2,Q) , so we can apply the general theory of such orders to study quadratic modules.
Note that the natural involution δ of the category of quadratic mod- ules that maps (M1, M2, H, P) to (M2, M1, P, H) corresponds to the following automorphism of the ring A:
a, b,
c1 2c2
c3 c4
7→
b, a,
c4 2c3
c2 c1
.
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This involution induces an involution on the category of right continu- ous quadratic functors, which we also denote by δ.
Here are some examples of quadratic functors and the corresponding A-modules, which play an important role in what follows:
Examples. (1) The tensor square ⊗2 : A 7→ A⊗A corresponds to the projective A-module A2 = Ae2. Its dual P2 =δ⊗2 is the so-called “quadratic construction” mapping a group A to I(A)/I3(A) , where I(A) is the augmentation ideal of the group ring Z[A] . It corresponds to the projective module A1 =Ae1. By the way, A1 and A2 are the only indecomposable projective A-modules.
(2) Denote by B1 the projection of A1 onto Z×0×0 , by B2 the projection of A2 onto 0×Z×0 , by B3 and B4, respectively, the projections of A1 and of A2 onto 0×0×Mat(2,Z) . Then B2 and B4 correspond, respectively, to the functors of theouter square V2
and of the symmetric square S2, while B1 and B3 correspond to their duals, δV2
= I being the identity functor and δS2 = Γ2 being the Whitehead functor, which represents the “universal quadratic function” [15].
Consider the subring B of Z×Z×Mat(2,Z) consisting of all triples
a, b,
c1 2c2 c3 c4
, and its ideal J consisting of the triples
2a,2b,
2c1 2c2 c3 2c4
.
This ideal is indeed the conductor of B in A, i.e., the biggest ideal of B contained in A. The ring B ishereditary(i.e., of the global ho- mological dimension 1). Hence, each finitely generated B-module is a direct sum of factor-modules N/N0, with N being an indecomposable projective B-module. In our case, N is isomorphic to one of the mod- ules Bj defined above. Moreover, every homomorphism g : Q → P of projective (finitely generated) B-modules is “diagonalizable,” i.e., there are decompositions of Q and P into direct sums of indecompos- able modules: Q=Lm
i=1Qi and P =Ln
j=1Pj, such that the matrix of g with respect to this decomposition is diagonal [10].
3. Localization
From now on all modules are supposedfinitely generated, so we omit these words everywhere. In terms of quadratic functors, we restrict
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ourselves by the functors fab→ab. For any abelian group (in partic- ular, for every A-module) M denote by tM its torsion part, i.e., the set of all elements of finite order, and by fM itstorsion free part i.e.
the factor M/tM.
For any prime p∈ N, denote by (p) the functor of the (complete) localization at p. It maps every abelian group A to A(p) =A⊗Zp, where Zp is the ring of p-adic integers. Evidently, A(p) =B(p) =Zp× Zp ×Mat(2,Zp) if p 6= 2 . Hence, in this case, every indecomposable A(p)-module is isomorphic either to one of the modules Bi(p) (i = 1,2,3 ) or to a factor Bi(p)/pkBi(p) for some k. (Note that, for p6= 2 , B3(p) 'B4(p).)
Put Q=Q×Q×Mat(2,Q) . For every A-module M, Q⊗AM is a module over the semi-simple algebra Q, hence, Q⊗AM 'L3
i=1riUi, where Ui are simple Q-modules: Ui =Q⊗ABi (note that U4 'U3).
Denote by r(M) the vector (r1, r2, r3) , which we call the rank vector of the module M. We use the same definitions and notations for A(p)- modules, certainly, replacing Q by the field Qp of p-adic numbers.
Theorem 3.1. (1) A-modules M, N are isomorphic if and only if M(p) 'N(p) for all prime p.
(2) Given any set {Np}, where Np is an A(p)-module, there is an A-module M such that M(p) ' Np for all p if and only if almost all (i.e., all but a finite set) of them are torsion free (maybe, zero) and r(Np) = r(Nq) for all p, q. In this case, r(M) =r(Np) and tM =L
ptNp.
Proof. The theorem is obvious if all considered modules are torsion.
Consider the case when these modules aretorsion free. Then the second claim of the theorem is well-known [14]. The first one is also obvious for torsion free B-modules, as they are direct sums of Bi and EndBi =Z (cf., e.g., [8]). Moreover, given any torsion free A-module M, the isomorphism classes of modules N such that N(p) ' M(p) for all p and B⊗A N ' B ⊗AM are in one-to-one correspondence with the double cosets
(1) Aut(B⊗AM)\Aut(B⊗AM(2))/AutM(2)
(cf. ibid.). But, certainly, HomA(M, M0)⊇HomB(B⊗AM, M0J) for every two A-modules M, M0 and all the factors
HomB(Bi,Bj)/HomB(Bi, JBj)
are zero for i 6= j and F2 for i = j. Hence, any automorphism of B⊗AM(2) is congruent to an automorphism of B⊗A M modulo
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AutM(2). Thus, the set (1) always consists of a unique element, so, the theorem is true for torsion free A-modules.
For any A-module M, one can consider the exact sequence 0−→tM −→M −→fM −→0
which defines an element ξM ∈Ext1A(fM,tM) . As there are no homo- morphisms from tM to fM, two modules N and M are isomorphic if and only if there are isomorphisms α : tM →tN and β : fM →fN such that αξM = ξNβ (we mean the Yoneda multiplication). Put F = fM and T = tM. Then Ext1A(F, T) ' ExtA(2)(F(2), T(2)) (as A(p) is hereditary for p6= 2 ). In particular, one only has to consider the case, when T is a 2-group, i.e., T =T(2). Thus, the second claim of the theorem becomes evident. Moreover, in this case the isomor- phism classes of the A-modules N such that N(p) ' M(p) for all p are in one-to-one correspondence with the double cosets
(2) AutF \AutF(2)/Aut0F(2),
where Aut0F(2) denotes the subgroup of all automorphism acting triv- ially on Ext1A(F, T) .
Now note that J(2) = radA(2) (Jacobson radical) and the ring A(2) is semi-perfect in the sense of [1]. Hence, there always is an exact sequence
0−→K −→P −→F(2) −→0,
where P is a projective A(2)-module and K ⊆ J P . It induces an exact sequence
0−→K −→J P −→J F(2) −→0.
As both J P and J F(2) are B-modules and B is hereditary, the latter splits. It implies that αξ = 0 for any endomorphism α : F(2) → F(2) whose image belongs to J F(2) and for any element ξ ∈ Ext1A(F, T) . Hence, Aut0F(2) contains the “congruence subgroup”
α∈AutF(2)| Im(α−id) ⊆J F(2) .
One can easily check that Bi are the only irreducible torsion free A-modules. As each of them is a factor-module of some Aj and the length of Q⊗AAj is 2 , A(2) is an order of weight 2 in the sense of [12]. Moreover, as JAj decomposes, it also follows from [12] that the modules Aj are the only indecomposable but not irreducible torsion free A-modules. Thus, every torsion free A-module is a direct sum of modules isomorphic to Bi or Aj. The next table describes the values
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of the groups HomA(2)(F(2)0 , F(2)00 )/HomA(2)(F(2)0 , J F(2)00 ) , when F0 and F00 run through indecomposable torsion free A-modules:
B1 B2 B3 B4 A1 A2 B1 F2 0 0 0 F2 0 B2 0 F2 0 0 0 F2
B3 0 0 F2 0 F2 0 B4 0 0 0 F2 0 F2 A1 0 0 0 0 F2 0 A2 0 0 0 0 0 F2
It evidently implies that every automorphism of F(2) is congruent to an automorphism of F modulo Aut0F(2), i.e., the set (2) consists of one element. Hence, the theorem is valid for all (finitely generated)
A-modules.
4. Reduction
From now on, except of the last section (“Corollaries”), we con- sider the 2-local case, hence, always write A, B, Q, etc. instead of A(2), B(2), Q(2), etc.
We shall studyprojective presentationsofA-modulesM, i.e., homo- morphisms of projective modules f :P0 →P such that M ' Cokerf.
In terms of quadratic functors, it means that we shall study exact se- quences
Φ0 →Φ→F →0,
where F is a given functor, while Φ and Φ0 are direct sums of func- tors isomorphic to ⊗2 and P2. Moreover, if we are studying finitely generated modules, these sums can always be chosen finite too. The advantage of the local case is that here A is asemi-perfect ring in the sense of [1]; hence, every (finitely generated) A-module M has apro- jective cover, i.e., an epimorphism ϕ:P →M, where P is projective and Kerϕ ⊆J P (note that here J = radA, the Jacobson radical of A). It implies that M has a projective presentation f :P0 →P with Imf ⊆J P and we only consider such presentations.
Each homomorphism of projective modules f :P0 → P induces the homomorphism B⊗f : B⊗A P0 → B⊗A P. If M = Cokerf, then Coker(B⊗ f) ' B ⊗A M. As we know all possibilities for B ⊗ f, we are going to classify the homomorphisms f with prescribedB⊗f, or, the same, the A-modules M with prescribed B⊗AM. We always identify P with the submodule 1⊗P ⊆ B⊗AP. If Imf ⊆ J P, then also Im(B⊗f)⊆J(B⊗AP) =J P. On the other hand, suppose that
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Imϕ⊆J P for someϕ:B⊗AP0 →B⊗AP. As HomB(B⊗AP0, J P)' HomA(P0, J P), then ϕ is of the formB⊗f for some f :P0 →P.
One calls two homomorphisms f, f0 : P0 → P equivalent if there are automorphisms g ∈ AutP and g0 ∈ AutP0 such that gf = f0g0. Then, of course, Cokerf 'Cokerf0. The following proposition is quite evident:
Proposition 4.1. LetP, P0 be projective (finitely generated)A-modules, Q = B⊗A P, Q0 = B ⊗A P0, ϕ : Q0 → Q a homomorphism with Imϕ⊆J Q. There is a one-to-one correspondence between the equiva- lence classes of homomorphisms f :P0 →P such that B⊗f =ϕ and the double cosets
(3) Autϕ\AutQ×AutQ0/AutP ×AutP0,
whereAutϕdenotes the subgroup{(g, h)∈AutQ×AutQ0|gϕ=ϕh}.
Namely, the coset containing a pair (g, h) corresponds to the homo- morphism f which coincides with g−1ϕh under the identification of HomA(P0, J P) and HomB(Q0, J Q) .
Put P = P/J P and Q = Q/J Q. Note that always J Q = J P and AutP contains the “congruence subgroup”
Aut(Q, J) ={g ∈AutQ| Im(g−id) ⊆J Q} .
Moreover, an endomorphism g of Q is invertible if and only if it is invertible modulo J. Therefore, the set (3) can be identified with (4) Autϕ\AutQ×AutQ0/AutP ×AutP0,
where Autϕ is the image of Autϕ in AutQ×AutQ0.
Every homomorphism ϕ:Q0 →Q of projective B-modules is equiv- alent to a direct sum of homomorphisms of the forms ϕijk : Bj → Bi. Here (ij)∈ {(11),(22),(33),(34),(43),(44)},k is a non-negative inte- ger or, if i=j, maybe k =±∞. If k 6=±∞, the homomorphism ϕijk is the multiplication by 2k+1 for i =j, the multiplication on the right by 2kh for (ij) = (34) and by 2kp for (ij) = (43); ϕii,−∞ denotes the homomorphism 0→Pi and ϕii,+∞ the homomorphism Pi →0.
The rings A/J and B/J are semi-simple; Ai (i= 1,2) and Bj (j = 1,2,3,4) are their simple modules. Moreover, B1 ' B3 ' A1 and B2 'B4 'A2 asA-modules, and the isomorphismsB⊗AA1 'B1⊕B3, B⊗AA2 'B2⊕B4 induce the diagonal embeddings
EndA1 'F2 −→ EndB1 ×EndB3 'F22, EndA2 'F2 −→ EndB2 ×EndB4 'F22,
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Forϕ :Q0 →Q,ψ :Q01 →Q1, one denotes:
Hom(ϕ, ψ) = {(u, v)|u:Q→Q1, v :Q0 →Q01, uϕ=ψv}, H(ϕ, ψ) and H0(ϕ, ψ) the projections of Hom(ϕ, ψ), respectively, onto HomB(Q, Q1) and HomB(Q0, Q01). One easily calculates these groups whenϕ, ψrun through{ϕijk}. It is convenient to present the result in the following way. Consider the new symbolsv(ijk) andv0(ijk), where the triples (ijk) are as before except v(ii,+∞) and v0(ii,−∞), which are forbidden. We define an ordering on the set of these symbols putting v(ijk) ≤ v(i0j0k0) if H(ϕ(ijk), ϕ(i0j0k0))6= 0 and v0(ijk) ≤ v0(i0j0k0) if H0(ϕ(ijk), ϕ(i0j0k0))6= 0 (then all these groups are isomorphic to F2).
Here is the table describing this ordering:
v(ijk)< v(ijk0) ifk0 < k ,
v(33k)< v(34k0) if k0 < k , otherwise v(34k0)< v(33k), v(44k)< v(43k0) if k0 ≤k , otherwise v(43k0)< v(44k), v0(ijk)< v0(ijk0) if k < k0,
v0(33k)< v0(43k0) ifk < k0, otherwise v0(43k0)< v0(33k), v0(44k)< v0(34k0) ifk ≤k0, otherwise v0(34k0)< v0(44k). Now one can easily see that we have obtained a special case of the so called “bunch of chains” (cf. [7] or Appendix). Namely, put I = {1,2,3,4,5,6,7,8}, consider the following chains:
E1 ={v(11k)} F1 ={1} E2 ={v(22k)} F2 ={2} E3 ={v(33k), v(34k)} F3 ={3} E4 ={v(43k), v(44k)} F4 ={4} E5 ={v0(11k)} F5 ={10} E6 ={v0(22k)} F6 ={20} E7 ={v0(33k), v0(43k)} F7 ={30} E8 ={v0(34k), v0(44k)} F8 ={40}
(with the above defined ordering) and define an equivalence relation ∼ on the union of these chains such that the only non-trivial equivalences are: v(ijk)∼v0(ijk) for k6=∞, 1∼3 , 2∼4 , 10 ∼30, 20 ∼40. One gets in this way a bunch of chainsX={I,Ei,Fi,∼ }. We are going to establish relations between representations of this bunch over the field F2 and quadratic modules. Thus, in all references to [7] or Appendix one supposes k = F2. We write x−y if x ∈ Ei, y ∈ Fi (with the same i) or vice versa and denote by U the bimodule corresponding to the bunch X (cf. Apendix).
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For every equivalence class a of the relation ∼ or, the same, for an object of the category C=C(X) (cf. Appendix), put
QE(a) =
(Bi if a3v(ijk) for some j, k 0 otherwise
Q0E(a) =
(Bj if a3v(ijk) for some i, k 0 otherwise
QF(a) =
(Bi⊕Bj if a ={i, j} ⊆ {1,2,3,4}
0 otherwise
Q0F(a) =
(Bi⊕Bj if a ={i0, j0} ⊆ {10,20,30,40}
0 otherwise
If c = ⊕mam is an object of the category C∞ (the additive hull of C), put QE(c) = L
mQE(am) , the same for Q0E(c), QF(c), Q0F(c) . Note that U(a, b) 6= 0 if and only if a ⊆ F, b ⊆ E and there are x∈a, y∈b such that x−y, in which case this space is 1-dimensional and can be identified with one of the spaces HomB(QF(a), QE(b)) or HomB(Q0F(a), Q0E(b)) , namely, the non-zero one. Hence, any el- ement u ∈ U(c, c) can be considered as a pair of homomorphisms (α(u), α0(u)) , where α(u) : QF(c) → QE(c) and α0(u) : Q0F(c) → Q0E(c) . Call such an element u (i.e., a representation of the bunch X) balanced if both α(u) and α0(u) are isomorphisms and there are such projective A-modules P, P0 that QF(c)'B⊗AP and Q0F(c)' B⊗AP0. Then the following result is immediate.
Theorem 4.2. There is a one-to-one correspondence between the equiv- alence classes of homomorphisms of projective A-modules f :P0 →P such that Imf ⊆ J P and the isomorphism classes of balanced repre- sentations of the bunch of chains X.
5. Description
Now we are able to use the results of [7] in order to describe the projective presentations of quadratic modules (or, the same, of A- modules). We rearrange a bit the definitions of [7] or Appendix in order to make them more adequate to the specific structure of our con- crete bunch of chains X as well as to the “balancedness condition” of Theorem 4.2. One writes x∈ X and says that x is an element of X if x is an element of one of the chains Ei or Fi. One also writes x−y if x∈Ei and y∈Fi with the same i or vice versa.
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Definition 5.1. (1) An X-word (or simply a word) is a finite se- quence w = x1r2x2r3. . . rnxn, where xi ∈ X and ri ∈ { −,∼ }, satisfying the following conditions:
(a) For every i= 2, . . . , n, xi−1rixi in the above defined sense.
(b) For everyi= 2, . . . , n−1, ri 6=ri+1.
(2) An X-word w = x1r2x2r3. . . rnxn is called balanced if n > 1 and both x1 and xn are of the form v(ii,−∞) or v0(ii,+∞) . (Note that in this case necessarily r2 = rn = − and n ≡ 0 (mod 4).) Moreover, if n = 4 , we suppose that either x1 or x4 is of the form v(ii,−∞) .
(3) AnX-word w=x1r2x2r3. . . rnxn is called cyclicif r2 =rn and xnr1x1, where r1 ∈ { ∼,− } and r1 6=rn. For such a word, we put x0 =xn and xn+1 =x1. (Note that in this case necessarily n≡0 (mod 8) .)
(4) A cyclicX-word is calledaperiodic if it cannot be obtained as a repetition of a shorter word: w6=wrwr . . . w for r ∈ { −,∼ }. (5) A cyclic word w = x1r2x2r3. . . rnxn is called normalized if it
is aperiodic, r2 =rn=∼ and x1 ∈Fi for some i.
(The last notion is not but a technical one making easier the definition of “bands” nearby.)
In [7] (cf. also Appendix) a complete description of the representa- tions of a bunch of chains was given in terms of the so calledstrings and bands. We are going to translate it to the description of A-modules via Theorem 4.2. To this purpose, we introduce some notions and notations.
Call an A-entry any word a of the form a =x ∼y, where x ∈Fi for some i and x 6= y. For every A-entry a = x ∼ y define two projective A-modules P(a) and P0(a) putting:
P(a) =
A1 if 1∈ {x, y} A2 if 2∈ {x, y} 0 otherwise (5)
P0(a) =
A1 if 10 ∈ {x, y} A2 if 20 ∈ {x, y} 0 otherwise (6)
For every word w, put P(w) = L
aP(a) and P0(w) = L
aP0(a) , where a runs through all subwords of w which are A-entries.
If a and a0 are two subwords of w which are both A-entries, one says that they arerelated in w by the triple (ijk) if they are contained in a subword of the form a−ω(a, a0)−a0, where ω(a, a0) =v(ijk)∼v0(ijk) ,
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or of the form a0−ω(a, a0)−a, where ω(a, a0) = v0(ijk)∼v(ijk) . Note that in both cases P(a0) = P0(a) = 0 , while P(a) and P0(a0) are non- zero. (In particular, this relation between a and a0 isnot symmetric.) We call ω(a, a0) the relating subword for a, a0. If w is a normalized cyclic word, we also include the case when a = xn−3 ∼ xn−2, a0 = x1 ∼x2 or vice versa and ω(a, a0) =xn−1 ∼xn.
Now define the homomorphisms θ(ijk) : Aj → Ai, where k >0 or k = 0 and i6=j. Namely, θ(ijk) is the multiplication to the right by the following element of the ring A:
2k+1,0, 0 0
0 0
if i=j = 1
0,2k+1, 0 0
0 0
if i=j = 2
0,0,
2k+1 0
0 0
if i=j = 3
0,0,
0 0 0 2k+1
if i=j = 4
0,0,
0 0 2k 0
if i= 3, j = 4
0,0,
0 2k+1
0 0
if i= 4, j = 3
Definition 5.2. Let w = x1r2x2r3. . . rnxn be a balanced X-word.
Then the corresponding string homomorphism c(w) : P0(w) → P(w) is defined as one whose only non-zero components with respect to the decompositions P(w) = L
aP(a) and P0(w) = L
aP0(a) are P0(a0) → P(a) if a and a0 are related in w by a triple (ijk) , the corresponding component being θ(ijk) .
The string module C(w) is defined as Cokerc(w) .
Note that the string homomorphism can be defined also when n = 4 and both x1 and x4 are of the form v0(ii,+∞) , but then it is the zero mapping Aj →0 , hence, gives rise to the zero module.
Definition 5.3. Let noww=x1r2x1r2. . . rnxn be a normalized cyclic word, π(t) 6= td be a primary polynomial over the filed F2, i.e., a power of an irreducible one, d = degπ(t) . Denote by Φ the Frobe- nius matrix with the characteristic polynomial π(t) and by I the unit d×d matrix. Then the band homomorphism b(w, π(t)) is de- fined as the mapping dP0(w) → dP(w) , whose only non-zero com- ponents with respect to the decompositions dP(w) = L
adP(a) and
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dP0(w) = L
adP0(a) are dP0(a0) → dP(a) if a and a0 are related in w by a triple (ijk) , the corresponding component being θ(ijk)I, except of the case when the relating subword ω(a, a0) is the subword xn−1 ∼ xn (hence, {a, a0} = {x1 ∼x2, xn−3 ∼xn−2}); in the latter case the corresponding component is θ(ijk)Φ .
The band module B(w, π(t)) is defined as Cokerb(w, π(t)) .
The following classification is an immediate consequence of that given in [7].
Theorem 5.4. (1) For every balanced wordw, the string A-module C(w) is indecomposable and for every pair (w, π(t)) with w a normalized cyclic word, π(t) 6= td a primary polynomial over F2, the band A-module B(w, π(t)) is also indecomposable.
(2) Every finitely generated A-module decomposes uniquely into a direct sum of modules isomorphic to string and band A-modules.
(3) The only isomorphisms between different string and band A- modules are the following:
(a) C(w) ' C(w∗), where w∗ is the reversed word to w, i.e., if w=x1r2x2r3. . . rnxn then w∗ =xnrn. . . r3x2r2x1. (b) B(w, π(t)) ' B(w(s), π(t)) where s is a positive integer
divisible by 4 and w(s) is the s-shift of w, i.e., if w = x1r2x2. . . rnxnthenw(s) =xs+1rs+2xs+2. . . rnxn−x1. . . rsxs. (The condition “ 4|s” is necessary in order that w(s) were also a normalizes cyclic word.)
(c) B(w, π(t))'B(w∗(s−2), π∗(t)) for s as above and π∗(t) = tdπ(1/t) where d= degπ(t).
6. Corollaries
We present some corollaries of the description of the A(2)-modules given above. First, note that one can also construct string and band modules for the ring A just in the same way as it has been done in Sec- tion 6 for its localization A(2). To be accurate, we will denote from now on the string and band A(2)-modules by C(2)(w) and B(2)(w, π(t)) , while the notations C(w) and B(w, π(t)) will be used for the string and band A-modules. Denote also by B(i, p, k) the factor-module Bi/pkBi, where i ∈ {1,2,3,4}, k ∈ N and p is an odd prime. As the “local” string and band modules are indeed the 2-localizations of the “global” ones, Theorems 3.1 and 5.4 imply the following complete description of finitely generated A-modules (or, the same, quadratic modules).
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Theorem 6.1. (1) For every balanced wordw, the string A-module C(w) is indecomposable and for every pair (w, π(t)) with w a normalized cyclic word, π(t) 6= td a primary polynomial over F2, the band A-module B(w, π(t)) is also indecomposable.
(2) Every finitely generated A-module decomposes uniquely into a direct sum of modules isomorphic to string and band A-modules and to modules B(i, p, k).
(3) The only isomorphisms between different string and band A- modules are the following:
(a) C(w)'C(w∗).
(b) B(w, π(t)) ' B(w(s), π(t)) ' B(w∗(s−2), π∗(t)) where s is a positive integer divisible by 4.
(4) Neither two different modules B(i, p, k) are isomorphic and nei- ther of them is isomorphic to a string or band module.
We are also able to write down generators and relations for in- decomposable quadratic modules. Namely, we say that a balanced word w = x1r2x2. . . rnxn, as well as the corresponding string mod- ule, is of type −∞ (resp., +∞ or ±∞) if both ends of w are of the form v(ii,−∞) (resp., both are v0(ii,+∞) or one is v(ii,−∞) and the other is v0(jj,+∞) ). Certainly, in the latter case (±∞), one can always suppose (and we shall do it) that x1 = v(ii,−∞) and xn = v0(jj,+∞) . Just in the same way, we shall suppose that a normalized cyclic word w starts from x1 ∈ {1,2,3,4}. Hence, such words are of the following forms:
• word of type ±∞:
v(i1i1,−∞)−i1 ∼i2−v(i2j1k1)∼v0(i2j1k1)−
−j10 ∼j20 −v0(i3j2k2)∼ · · · ∼v0(i2mj2m−1k2m−1)−
−j2m−10 ∼j2m0 −v0(j2mj2m,+∞)
• word of type −∞:
v(i1i1,−∞)−i1 ∼i2−v(i2j1k1)∼v0(i2j1k1)−
−j10 ∼j20 −v0(i3j2k2)∼ · · · ∼v(i2m−1j2m−2k2m−2)−
−i2m−1 ∼i2m−v(i2mi2m,−∞)
• word of type +∞:
v0(j−1j−1,+∞)−j−10 ∼j00 −v0(i1j0k0)∼v(i1j0k0)−
−i1 ∼i2−v(i2j1k1)∼ · · · ∼v0(i2mj2m−1k2m−1)−
−j2m−10 ∼j2m0 −v0(j2mj2m,+∞)
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• normalized cyclic word:
i1 ∼i2−v(i2j1k1)∼v0(i2j1k1)−j10 ∼j20−
−v0(i3j2k2)∼ · · · ∼v0(i2m−1j2m−2k2m−2)−j2m−10 ∼
∼j2m0 −v0(i1j2mk2m)∼v(i1j2mk2m) Define the following operators acting on any quadratic module M:
R11= 2−P H, R22= 2−HP,
R33=P H, R44 =HP, R34=H, R43=P .
Here R11 and R33 are mappings M1 →M1, R22 and R44 are map- pings M2 → M2, while R34 : M2 → M1 and R43 : M1 → M2. (Indeed, Rij corresponds to the multiplication by θ(ij0) .)
Corollary 6.2. (1) If an indecomposable quadratic module M cor- responds to one of the balanced words described above, it is gen- erated by the elements gν (ν = 1,2, . . . , m) such that gν ∈M1 if i2ν ∈ {1,3} and gν ∈M2 if i2ν ∈ {2,4}. These elements are subject to the following defining relations:
(7) 2k2ν−1Ri2νj2ν−1gν + 2k2νRi2ν+1j2νgν+1 = 0. (We put g0 =gm+1 = 0.)
(2) If an indecomposable quadratic module M corresponds to a nor- malized cyclic word described above and to a primary polynomial π(t) = td+a1td−1 +· · ·+ad (aµ ∈ {0,1}), it is generated by the elements gνµ (ν = 1,2, . . . , m, µ = 1,2, . . . , d) such that gνµ ∈ M1 if i2ν ∈ {1,3} and gνµ ∈ M2 if i2ν ∈ {2,4}. These elements are subject to the following defining relations:
2k2ν−1Ri2νj2ν−1gνµ+ 2k2νRi2ν+1j2νgν+1,µ = 0 if ν < m , 2k2m−1Ri2mj2m−1gmµ+ 2k2mRi1j2mg1,µ+1 = 0 if µ < d , 2k2m−1Ri2mj2m−1gmd−2k2mRi1j2m
d
X
µ=1
aµg1µ= 0. (8)
Remark. Certainly, the generators of M1 as of abelian group are, for string modules:
{gν, phgν|gν ∈M1} ∪ {pgν|gν ∈M2} , while for band modules:
{gνµ, phgνµ|gνµ∈M1} ∪ {pgνµ|gνµ∈M2} and those of M2 are, for string modules:
{gν, hpgν|gν ∈M2} ∪ {hgν|gν ∈M1} ,
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while for band modules:
{gνµ, hpgνµ|gνµ∈M2} ∪ {hgνµ|gνµ∈M1} .
One can easily deduce the defining relations for these elements from the relations (7),(8): one only has to multiply each of them by h and ph or by p and hp.
Evidently, the projective indecomposable A-modules A1 and A2 correspond to the words, respectively, v(11,−∞)−1∼3−v(33,−∞) and v(22,−∞)−2∼4−v(44,−∞) (the only balanced words of length 4). The description given by Theorem 6.1 allows easily to calculate pro- jective resolutions of quadratic modules. First, it implies immediately the following result.
Proposition 6.3. The kernels of string and band morphisms are the following:
• Kerb(w, π(t)) = 0;
• Kerc(w) = 0 if both ends of the word w are of the form v(ii,−∞);
• Kerc(w) ' Bλ(i) if one end of the word w is v0(ii,+∞) and the other is v(jj,−∞);
• Kerc(w)'Bλ(i)⊕Bλ(j) if both ends of w are v0(ii,+∞) and v0(jj,+∞),
where λ(1) = 1, λ(2) = 2, λ(3) = 4, λ(4) = 3.
Now note that each module Bi has a periodic projective resolution of period 4 . Here they are:
. . .−→A2 −→α1 A1 −→α4 A1 −→α3 A2 −→α2 A2 −→α1 A1 −→B1 . . .−→A1 −→α3 A2 −→α2 A2 −→α1 A1 −→α4 A1 −→α3 A2 −→B2 . . .−→A1 −→α4 A1 −→α3 A2 −→α2 A2 −→α1 A1 −→α4 A1 −→B3 . . .−→A2 −→α2 A2 −→α1 A1 −→α4 A1 −→α3 A2 −→α2 A2 −→B4 where α1 =θ(341) , α2 =θ(221) , α3 =θ(431) , α4 =θ(111) .
Corollary 6.4. (1) A quadratic module M is of finite projective dimension if and only if it contains no string summands of type +∞ or ±∞. In this case either M is projective (hence, a direct sum of modules isomorphic to Ai) or pr.dimM = 1. Otherwise, pr.dimM =∞.
(2) Any quadratic module M has a periodic projective resolution
· · · →Pn+1 −→γn Pn −−−→γn−1 . . .−→γ1 P1 −→γ0 P0 →M →0 of period 4, namely, such that γn+4 =γn for every n ≥2.
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In particular, we obtain the value of the finitistic dimension [1] of the ring A or, the same, of the category of quadratic modules.
Corollary 6.5. fin.dimA = 1.
We are also able to calculate the torsion free part fM of every indecomposable quadratic A-module M.
Corollary 6.6. The torsion free parts of non-projective string and band modules are the following:
• fB(w, π(t)) = 0;
• fC(w) = 0 if both ends of w are of the form v0(ii,+∞);
• fC(w)'Bi if one of the ends of w is v(ii,−∞) and the other is v0(jj,+∞);
• fC(w)'Bi⊕Bj if both ends of w are v(ii,−∞) and v(jj,−∞) and w is not of length 4.
Appendix: Bunches of chains
Here we recall some definitions and results related to thebunches of chains considered by Bondarenko in [7]. We rearrange the definitions to make them more convenient for our purpose and consider only the case of chains (not semi-chains) as we need only this one and it is technically much easier. In what follows k denotes an arbitrary field.
Definition A.7. A bunch of chains X ={I,Ei,Fi,∼ } is defined by the following data:
(1) A set I of indexes.
(2) Two chains (i.e., linear ordered sets) Ei and Fi given for each i∈I.
Put E:=S
i∈IEi, F:=S
i∈IFi and |X|:=E∪F.
(3) An equivalence relation ∼ on |X| such that each equivalence class consists of at most 2 elements.
We also write x−y if x ∈ Ei, y ∈ Fi or vice versa (with the same index i). Moreover, we consider the ordering on |X| which is just the union of all orderings on Ei and Fi (i.e., x < y means that x, y belong to the same chain Ei or Fi and x < y in this chain).
If a bunch of chains X = {I,Ei,Fi,∼ } is given, define the corre- sponding k-category C = C(X) and the corresponding C-bimodule U=U(X) as follows:
• The objects of C are the equivalence classes of |X| with respect to ∼.
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• If a, b are two such equivalence classes, a basis of the morphism space C(a, b) consists of elements pxy with y∈a, x∈b, y <
x and, if a=b, the identity morphism 1a.
• The multiplication is given by the rule: pxypyz = pxz if z <
y < x, while all other possible products are zeros.
• A basis of U(a, b) consists of elements uxy with x ∈ b∩E, y∈a∩F and x−y.
• The action of C on U is given by the rule: pzxuxy = uzy if x < z; uxypyt =uxt if t < y, while all other possible products are zeros.
We also consider theadditive hull C∞ of the category C and the nat- ural prolongation of the bimodule U onto C∞, which we also denote by U.
Thecategory of representations of the bunch X is then defined as the category El(U) of the elements of this bimodule [9]. In other words, the objects of El(U) are the elements of S
AEl(A, A) where A runs through the objects of C∞. A morphism u→u0, where u∈El(A, A) , u0 ∈El(A0, A0) , is a morphism α :A →A0 such that αu =u0α. One can easily verify that this definition gives just the same representations as the definition from [7]. Note that in [7] a more general situation was investigated, but we only need this case, which is essentially simpler than the general one. The following result is the specialization of the description of the representations given in [7] to our case, though it can be obtained directly using the same recursive procedure. First define some combinatorial objects called “strings” and “bands.”
Definition A.8. Let X={I,Ei,Fi,∼ } be a bunch of chains.
(1) An X-word is a sequence w=x1r1x2r3x3. . . rmxm, where xk ∈
|X| and each rk is either ∼ or −, such that:
(a) xk−1rkxk in |X|;
(b) if rk =∼, then rk+1 =− and vice versa.
Possibly m = 1 , i.e., w=x for some x∈ |X|.
(2) Call an X-word w as above an X-cycle if r2 = rm =∼ and xm−x1. (Note that in this case m is always even.)
(3) Call an X-word fullif, whenever x1 is not a unique element in its equivalence class, then r2 =∼ and, whenever xm is not a unique element in its equivalence class, then rm =∼.
(4) Call an X-cycle w= x1r2x2. . . rmxm aperiodic if it cannot be written as a repetition of a shorter cycle v: w6=vrvr . . . v for any r∈ { −,∼ }.
(5) We say that an equivalence class a occurs in a word w if w contains a subword x in case a ={x} is a singleton, or either
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a subword x ∼ y or a subword y ∼ x in case a = {x, y} with x 6= y. In the former case we say that this occurrence corresponds to the occurrence of x, while in the latter case we say that it corresponds to both the occurrence of x and to the occurrence of y. Denote by ν(a, w) the number of occurrences of a in w.
Definition A.9. For an X-word w = x1r2x2. . . rmxm call its ∼- subword any subword of the form v =x∼y as well as that of the form v = x if x ∈ X is unique in its equivalence class. In the latter case put |v|={x}, while in the former case put |v|={x, y}. Denote by [w] the collection of all ∼-subwords of w.
Note that if w is a cycle, it contains no entries x ∈ |X| such that x is unique in its equivalence class.
Definition A.10. For any full X-word w = x1r2x2. . . rmxm, define the corresponding string representation u=us(w) of the bunch X as follows:
(1) u∈U(A, A) where A=L
v∈[w]|v|.
(2) Suppose there is a subword v1 −v2 in w with vi ∈ [w] . Let x be the right end of the word v1 and y be the left end of the word v2. Then U(A, A) has a direct summand U(|v1|,|v2|)⊕ U(|v2|,|v1|) and we define the corresponding components of u to be (0, uxy) if x∈E and (uyx,0) if x∈F.
(3) All other components of u are defined to be zero.
Definition A.11. For any pair (w, π(t)) where w is an aperiodic X- cycle and π(t) 6= td is a primary polynomial over k (i.e., a power of an irreducible one), define the corresponding band representation u=ub(w, π(t)) of the bunch X as follows:
(1) u∈U(A, A) where A=L
v∈[w]d|v| and d= degπ(t) .
(2) Suppose there is a subword v1 −v2 in w with vi ∈ [w] . Let x be the right end of the word v1 and y be the left end of the word v2. ThenU(A, A) has a direct summand
U(d|v1|, d|v2|)⊕U(d|v2|, d|v1|)'
Mat(d×d,U(|v1|,|v2|)⊕Mat(d×d,U(|v2|,|v1|)))
and we define the corresponding components of u to be (0, uxyI) if x∈E and (uyxI,0) if x∈F, where I denotes the unit d×d matrix.
(3) Let now v1 be the last and v2 be the first ∼-subword in w (they may coincide), x be the right end of the word v1 and
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y be the left end of the word v2. Then U(A, A) has a direct summand
U(d|v1|, d|v2|)⊕U(d|v2|, d|v1|)'
Mat(d×d,U(|v1|,|v2|)⊕Mat(d×d,U(|v2|,|v1|)))
and we define the corresponding components of u to be (0, uxyJ) if x∈E and (uyxΦ,0) if x∈F, where Φ denotes the Frobe- nius matrix with the characteristic polynomial π(t) .
(4) All other components of u are defined to be zero.
Theorem A.12. (1) All representations us(w) and ub(w, π(t)) de- fined above are indecomposable and each indecomposable repre- sentation of X is isomorphic to one of these representations.
(2) The only possible isomorphisms between these representations are the following:
(a) us(w) ' us(w∗) if w = x1r2x2. . . rmxm and w∗ = xmrm xm−1. . . r1x0, the reversed word.
(b) ub(w, π(t)) ' ub(w0, π0(t)) if w = x1r2x2. . . rmxm, w0 = x2k+1r2k+2x2k+2. . . r2kx2k is a cyclic permutation of wand π0(t) = π(t) for k even, while for k odd π0(t) = tdπ(1/t). (c) ub(w, π(t)) ' ub(w0, π0(t)) if w = x1r2x2. . . rmxm, w0 = x2kr2kx2k−1. . . r2k+2x2k+1 is a cyclic permutation of the re- versed word and π0(t) = π(t) for k odd, while for k even π0(t) = tdπ(1/t).
(d always denotes degπ.)
References
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